Practically, a capo will shift each string by a half tone per position, but what does it do to the intervals they produce with each other?
My shot is that e.g.
* without capo if we fret the simplest chord (about the only one I can play), Em = E2 B2 E3 G3 B3 E4, E-G-B with some harmonics,
* and we then transpose one halftone, (capo 1, fret G and B strings on third fret), we'll have F2 C3 F3 G#3 C4 F4. Still a minor chord, with an F fundamental, but I have to admit Wikipedia says Fm is actually not F-G#-C, but F-Ab-C ( https://en.wikipedia.org/wiki/Minor_chord ). Good enough ?

So a few questions:

Was I lucky in my example chord choice or does it always work like that?

If there IS a difference, is it all due to temperament?

If it sounds different (relatively, the halftone shift not considered), is it due to temperament, or to imperfect manual tuning?

What is actually the temperament when tuning with a spectrum analyzer app? with relative tuning ("5-5-5-4-5")?

A capo does nothing to the intervals -- that's the entire point, being able to shift everything up equally. Adding exactly n semitones always adds exactly n semitones, why do you think it possible for it to be inconsistent?
– Matthew Read♦Oct 21 '15 at 1:53

5 Answers
5

First of all, putting a capo on does not change the temperament of your guitar. Your guitar temperament is equal temperament so and that's not changing so forget about the temperament aspect of this.. The only thing it does is change what strings are "Open".

So without a capo, your open strings are the typical E-A-D-G-B-E. For every fret you move it, all your open strings go up one fret so putting it on the first fret would give you the notes F-A#-D#-G#-C-F. So now if you have the same chord shape it will be the same type of chord, but have a different root.

You can also think of by the typical chord diagrams. This is an Em chord:

Em

This is an Fm chord:

Fm

This is an F#m chord:

F#m

See how the F#m chord is one just 1 fret up from the Fm chord and the Fm chord is one just 1 fret up from the Em chord ? This is no coincidence since every minor chord has the same formula you can use the same shape to make the same type of chord.

For practical musical performance purposes, a capo simply transposes up by a semitone per fret. All notes are "theoretically" simply higher by the number of semitones times the number of frets you move the capo. In a perfect world, what the capo attempts to do is the same effect as re-tuning your guitar one half step sharp for each capo position.

In the real world, due to intonation limitations of guitar design, fret height variances, capo tension differences (from one capo to another) capo placement (how close to the fret), string gauge, scale length, action and set up, compensation in the bridge and other variables - the capo is not sophisticated enough to exactly replicate the change in tone the way it is envisioned. A capo is simply a mechanical device of varying designs applied to a guitar necks and strings of varying shapes and designs and tensions and set up variables.

So the best you can hope to accomplish is a close approximation. If you start measuring the effects with a spectrum analyzer app, you will be all over the board in varying directions depending on the type capo you are using and the guitar you are using it on and the set up on that guitar and the strings on that guitar.

Some guitar effects and virtual guitars have attempted to create "virtual capos" using digital effects and electronics to process and manipulate the signal. I have not had a chance to test these, but theoretically (there's that word again) they could potentially be more consistent and come closer to accomplishing what a capo attempts to do mechanically.

A capo is a transposition device: everything gets moved to a higher pitch while retaining the same voicing that you have in the lower position. The resulting chord voicing may differ from playing the chord with the same "name" without capo.

As an example, playing G major without capo will usually be voiced as
G B d g b g', so 1-3-5-8-10-15 in scale steps. Playing G major with a capo on third fret acquires the different chord voicing G d g b d' g', so it's rather 1-5-8-10-12-15 in scale steps like you'd use for playing E major without capo.

•Was I lucky in my example chord choice or does it always work like that?

It will always work like that. (Capo-ing 1 fret higher will result in the same chord quality (minor, major etc) with a name (in this case E) that is a half step up in the case of 1 fret since 1 fret = one half step.)

•If there IS a difference, is it all due to temperament?

What do you mean by temperament? As stated by others, that will not change directly. The difference is in pitch (what notes are played).

•If it sounds different (relatively, the halftone shift not considered), is it due to temperament, or to imperfect manual tuning?

It is different with regards to pitch. Temperament may mean something different than you think.

•What is actually the temperament when tuning with a spectrum analyzer app? with relative tuning ("5-5-5-4-5")?

I believe you are asking something different than you think here. Tuning is not the same as temperament. Temperament is to instrument construction as Tuning is to instrument use. (There are a few rare exceptions where temperament can be effected by tuning such as the piano.) Here is a good article about temperament. Look for the chapter on Fretted Instruments.

Note that @rockinCowboy, @user24218 and @Dom are correct without qualification.

Note that the act of placing a capo will stretch the strings, and you might stretch them unevenly, or cause a tension imbalance between the string on either side of the capo and nut. Either could introduce a need for retuning. I usually can place a capo without major complications in this direction (in other words I don't often need to retune), but there is always at least an imperceptible change in tension and thus tuning.

Perhaps you do know what temperament is: Well there is no such thing as a normal guitar with even temperament... but you can get closer if you have a fanned fret guitar:

The word Temperament.can be used in regards to using a kapo but not in the sence that you suggest.depending on the fret location of the kapo. It can enable a vocalist to sing lower or higher or you can sing a song in the master key and change the chord pattern completely .to my understanding using the word temperament in relation to any musicial devise is simply does it stay in tune .by placing a kapo on a guitar neck and it goes out of tune doesn't mean its temperamental it simply means its out of tune but if it goes out of tune the same way every time then one could say that your guitar is temperamental. Although a neck or fret adjustment could correct the problem.if your guitar tunes correctly without a kapo and the tonation is correct all the way up the neck. but will not tune with a kapo then you need a new guitar .its not temperamental its junk

Suggest you are struggling with translation here. Temperament and temperamentAL are completely different in this situation, and a capo has no bearing on temperament - or temperamental.
– TimJul 21 '16 at 11:00